How do I apply Euler' theorem on a nonplanar graph?

Question

How do I apply Euler's theorem (faces+vertices-edges=2) to a nonplanar graph to prove that it is nonplanar?

Answer 1

Euler's characteristic is a function of the genus.

$$\chi=2-2g$$

Roughly speaking, the genus is the number of holes in a shape.

For example, for a plane, the genus is 0 (no holes), and thus $\chi_{plane}=2$

And any planar graph must satisfy $$|V|-|E|+|F|=\chi_{plane}=2$$

Lets take for example $k_5$ (the complete graph on 5 vertices) which is not planar.

But, we can draw it on a torus like this:

$$|V|-|E|+|F|=5-10+5=0$$

Which is exactly $\chi_{torus}=2-2*1=0$